The quantum mechanical time evolution operator governs how observables and/or states evolve during finite time steps, and is always unitary. Use this tag for questions about the time evolution operator, or the different equations of motion in the Schrödinger/Heisenberg/Dirac pictures. For time-independent Hamiltonians, the time evolution operator is simply exp(-iHt).
Time-evolution of an observer means how it changes over time; it's derivative over time. In quantum mechanics, it is described by the Hamiltonian, i.e.
This is also related to Noether's theorem, which proves that time-invariance is equivalent to energy-conservation.
DO NOT use this tag for evolving laws of physics.